Prerequisites:
Functional Analysis A (Math 362A) or equivalent. If you plan to take
this course without having taken Math 362A in fall, please get the notes
from one of the students in Math 362A and work through them before
taking Math 362B.
Recommended Books:
There will be no textbook. The following books
contain part of what I plan to cover in the course:
1) John B. Conway, A Course in Functional Analysis,
Springer GTM 96, 2nd edition (January 1997).
2) Gert Pedersen, Analysis Now,
Springer Verlag, GTM 118, 1988 (revised edition).
3) Kehe Zhu, An Introduction to Operator Algebras,
CRC Press, 1993.
Additional references will be provided during the course.
Syllabus:
This course is a continuation of Math 362A (Functional Analysis A),
which I taught in the fall semester 2003. In the first part of
the course I will present trace class and Hilbert Schmidt operators
and then I will discuss abstract spectral theory in Banach algebras,
the Gelfand transform, the spectral theorem, continuous and Borel
functional calculus and all that. Some applications to quantum physics
will be presented.
In the second part of the course I will discuss a special topic
from functional analysis. This could be a topic from operator
theory (e.g. unbounded operators), operator algebras (e.g. von Neumann
factors), ergodic theory (e.g. invariants for measurable equivalence
relations), harmonic analysis on groups or representation theory (e.g.
amenability, Haagerup's property, property T, operator algebras
associated to groups etc.). The choice of the topic will depend on time
and interest of the audience.
Grading:
There will be no exams. The course grade will be based on
attendance, a presentation and homework problems.
Presentations:
1) Topological groups, existence and uniqueness of Haar measure
(Casey, Max).
2) Sobolev spaces, Sobolev embedding theorem, Rellich's theorem,
Elliptic Regularity theorem (David, Hannah, Yuliya).
3) The Dixmier trace: existence, properties, applications (Connes'
trace theorem) (Fumiko, Lin).
The presentations will be held on Mondays, 3-4pm, in SC 1404.